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International Journal for Multiscale Computational Engineering
Jacob Fish (open in a new tab) Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, New York 10027, USA
J. Tinsley Oden (open in a new tab) Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA
Somnath Ghosh (open in a new tab) Departments of Civil & Systems Engineering, Mechanical Engineering, and Material Science Engineering, Johns Hopkins University, Baltimore, MD, USA
Arif Masud (open in a new tab) Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 3129E Newmark Civil Engineering Laboratory, MC-250, Urbana, Illinois 61801-2352, USA
Klaus Hackl (open in a new tab) Institute of Mechanics of Materials, Ruhr-University Bochum, Bochum, 44721, Germany
Karel Matous (open in a new tab) Department of Aerospace and Mechanical Engineering, Center for Shock Wave-Processing of Advanced Reactive Materials, University of Notre Dame, Notre Dame, Indiana 46556, USA
Thomas J.R. Hughes (open in a new tab) Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, 201 East 24th Street, C0200, Austin, TX 78712-1229, USA
Caglar Oskay (open in a new tab) Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, Tennessee 37235, USA
Tamar Schlick (open in a new tab) Department of Chemistry, New York University, New York, New York 10003, USA; Courant Institute of Mathematical Sciences, New York University, New York, New York, 10012, USA; NYU-ECNU Center for Computational Chemistry, NYU Shanghai, China
The Impact Factor measures the average number of citations received in a particular year by papers published in the journal during the two preceding years. 2017 Journal Citation Reports (Clarivate Analytics, 2018) IF: 1.4 To calculate the five year Impact Factor, citations are counted in 2017 to the previous five years and divided by the source items published in the previous five years. 2017 Journal Citation Reports (Clarivate Analytics, 2018) 5-Year IF: 1.3 The Immediacy Index is the average number of times an article is cited in the year it is published. The journal Immediacy Index indicates how quickly articles in a journal are cited. Immediacy Index: 2.2 The Eigenfactor score, developed by Jevin West and Carl Bergstrom at the University of Washington, is a rating of the total importance of a scientific journal. Journals are rated according to the number of incoming citations, with citations from highly ranked journals weighted to make a larger contribution to the eigenfactor than those from poorly ranked journals. Eigenfactor: 0.00034 The Journal Citation Indicator (JCI) is a single measurement of the field-normalized citation impact of journals in the Web of Science Core Collection across disciplines. The key words here are that the metric is normalized and cross-disciplinary. JCI: 0.46 SJR: 0.333 SNIP: 0.606 CiteScore™:: 3.1 H-Index: 31

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ESSENTIAL FEATURES OF FINE SCALE BOUNDARY CONDITIONS FOR SECOND GRADIENT MULTISCALE HOMOGENIZATION OF STATISTICAL VOLUME ELEMENTS

pages 461-486
DOI: 10.1615/IntJMultCompEng.2012002929
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ABSTRACT

A second gradient continuum description developed, for example, by Germain, Toupin and Mindlin, and Eringen, gives rise to strain gradient plasticity, and is becoming a common coarse scale basis for multiscale homogenization of material response that respects the non-local nature of heterogeneous fine scale material response. Such homogenization approaches are developed to build either concurrent or hierarchical multiscale computational models for the second gradient response at the coarse scale that represent salient aspects of material response at the fine scale. Typically, the homogenization procedure consists of solving an initial boundary value problem for a statistical volume element of heterogeneous material at the fine scale and computing coarse scale stresses and strains using various volume averaging procedures. By enforcing a kinematically consistent description of the deformation field at each scale and asserting invariance of linear momentum with respect to scale of observation of a fixed set of mass particles, critical features of the boundary conditions and computation of homogenized stresses are revealed. In particular, an internal constraint on the higher-order fluctuation field is required to ensure orthogonality between that part of the fine scale deformation attributed to the second gradient and the part associated with higher-order fluctuations. Additionally, the body forces resulting from such internal constraints must be included in the computation of coarse scale stresses to respect scale invariance of linear momentum at each scale. Numerical implementation of fine scale fluctuation constraints employs linear constraint equations; the computation of coarse scale stresses is facilitated through a multiscale statement of principle of virtual velocities. Example fine scale simulations and associated coarse scale homogenization are presented to illustrate aspects of the boundary conditions.

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